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Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0,ω1])

Research output: Contribution to journalJournal article

Published

Journal publication date1/06/2012
JournalJournal of Functional Analysis
Volume262
Number of pages20
Pages4831-4850
Original languageEnglish

Abstract

Let ω1 be the smallest uncountable ordinal. By a result of Rudin, bounded operators on the Banach space C([0,ω1) have a natural representation as [0,ω1]×[0,ω1]-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on [0,ω1] defines a maximal ideal of codimension one in the Banach algebra
B(C([0,ω1])) of bounded operators on C([0,ω1]). We give a coordinate-free characterization of this ideal and deduce from it that B(C([0,ω1])) contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Functional Analysis 262 (11), 2012, © ELSEVIER.